Learn how to graph a quadratic

The first step is to determine the axis of symmetry using the formula x = -b / 2a. For the given equation f(x) = x^2 - 8x + 15, a = 1 and b = -8. Plugging these values in, x = -(-8) / (2 * 1) = 8 / 2 = 4. The axis of symmetry is the vertical line x = 4. This line divides the parabola into two symmetrical halves.

Next, determine if the graph opens up or down. This is based on the value of 'a'. If 'a' > 0, the parabola opens up, and if 'a' < 0, it opens down. In this case, a = 1, which is greater than 0, so the parabola opens upwards. This also means the vertex will be a minimum point.

The vertex always lies on the axis of symmetry. Since the axis of symmetry is x = 4, the x-coordinate of the vertex is 4. To find the y-coordinate (or f(x) value), plug x = 4 back into the original function: f(4) = 4^2 - 8(4) + 15 = 16 - 32 + 15 = -1. So, the vertex is at (4, -1).

To graph the parabola, choose two x-values to the left of the vertex (e.g., 3 and 2). Calculate their corresponding f(x) values: f(3) = 3^2 - 8(3) + 15 = 9 - 24 + 15 = 0, so (3, 0). f(2) = 2^2 - 8(2) + 15 = 4 - 16 + 15 = 3, so (2, 3). Due to the symmetry of the parabola around the axis x = 4, points at an equal distance to the right of the axis will have the same y-values. Therefore, for x = 5 (one unit to the right of 4, like 3 is one unit to the left), f(5) will be 0. For x = 6 (two units to the right of 4, like 2 is two units to the left), f(6) will be 3. This gives points (5, 0) and (6, 3), allowing for the complete sketching of the parabola.